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Scaling limits for interacting diffusions. (English) Zbl 0725.60085

The article studies the hydrodynamic limit, as \(N\to \infty\), of a system of N particles diffusing on a circle S and drifting under the influence of a potential arising from their pairwise interactions. The pair potential between a particle at x and a particle at y is of the form V(N(x-y)), where V has bounded support and is such that the interaction is repulsive. The author considers the evolution in time of the empirical measure of particle positions and shows that, under explicitly stated conditions, the distribution of the process of empirical measures converges weakly, as \(N\to \infty\), to a degenerate distribution concentrated on a single trajectory in the space of probability measures on S. At time t the trajectory is at a probability measure with density \(\rho\) (\(\theta\),t), \(\theta\in S\), where \(\rho\) (\(\theta\),t) is a solution of the nonlinear bulk diffusion equation \[ \partial \rho (\theta,t)/\partial t=[P(\rho (\theta,t))]_{\theta \theta}, \] P being the “pressure” of the corresponding grand canonical thermodynamic system on R. The method of proof is similar to the one used by M. Z. Guo, G. C. Papanicolaou and the author [ibid. 118, No.1, 31-59 (1988; Zbl 0652.60107)].

MSC:

60J60 Diffusion processes
60K35 Interacting random processes; statistical mechanics type models; percolation theory

Citations:

Zbl 0652.60107
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References:

[1] Guo, M.Z., Papanicolaou, G.C.: Self diffusion of interacting Brownian particles. Probabilistic Methods in Mathematical Physics, pp. 113–151, Ito, K., Ikeda N. (eds.) Tokyo: Kinokuniya 1987 · Zbl 0656.60109
[2] Guo, M.Z., Papanicolaou, G.C., Varadhan, S.R.S.: Nonlinear diffusion limit for a system with nearest neighbor interaction. Commun. Math. Phys.118, 31–59 (1988) · Zbl 0652.60107
[3] Spohn, H.: Equilibrium fluctuations for interacting Brownian particles. Commun. Math. Phys.103, 1–33 (1986) · Zbl 0605.60092
[4] Olla, S., Varadhan, S.R.S.: Scaling limit for interacting Ornstein-Uhlenbeck Processes. Commun. Math. Phys.135, 355–378 (1991) · Zbl 0725.60086
[5] Ruelle, David: Statistical mechanics; rigorous results. New York: Benjamin 1969 · Zbl 0177.57301
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